Converting Between Probabilities, Odds, Log-Odds, Exponentials, and Logarithms

From a probability to odds

Let

denote the probability of an event—for example, the probability that an eligible trial workspace converts to paid.

A probability is bounded:

If the conversion probability is $p=0.20$, then the probability of non-conversion is

Odds compare the probability of the event to the probability that it does not occur:

For $p=0.20$,

Thus the odds are $0.25$, often written as 1 to 4 against conversion: for every one conversion, there are four non-conversions in the corresponding ratio.

Probability and odds are related, but they are not interchangeable.

The central reference point is:

At a 50% probability, the event and non-event are equally likely. Below 50%, odds are below 1; above 50%, odds are above 1.

To reverse the conversion, begin with odds $o$:

Solving for $p$:

So the two directions are:

For instance, odds of $4$ imply

or an 80% probability.

Why take the logarithm of odds?

Odds are always positive:

That is an improvement over probabilities, but the scale remains asymmetric around the neutral point $1$. For example:

• odds of $1/4$ mean the event is disfavored;
• odds of $4$ mean the event is favored.

These are reciprocal quantities, but they do not look equally far from $1$ on the raw odds scale.

Taking the natural logarithm fixes this. The log-odds, also called the logit, are

The notation $\ln$ means the natural logarithm: logarithm with base $e$, where

The logarithm and exponential functions undo each other:

provided $a>0$.

Here are the same probabilities on all three scales:

Notice the symmetry:

The neutral probability $p=0.5$ corresponds to odds of $1$, and therefore to log-odds of zero:

Unlike probability and odds, log-odds can take any real value:

This is why log-odds are useful for a linear model. A weighted sum such as

can be any real number. Logistic regression treats this linear score $z$ as log-odds, not as a probability.

A crucial distinction:

The first expression is the log of a probability. The second is the logit, or log-odds. In logistic regression, it is the second quantity that is modeled as linear.

Returning from log-odds to probability: exponentials and the sigmoid

Suppose a model or calculation gives a log-odds value $z$:

Exponentiate both sides:

The left-hand side is now the odds. Solving for $p$ gives:

An equivalent and computationally familiar form is:

This function is called the sigmoid or logistic function:

It transforms every finite real number into a number strictly between $0$ and $1$:

The image shows three especially useful landmarks:

The sigmoid is not “making the relationship linear.” It does the opposite: it turns a linear log-odds score into a nonlinear probability curve. Near $z=0$, changes in the score can visibly change probability. At very large positive or negative scores, probabilities saturate near $1$ or $0$.

Odds ratios: the multiplicative meaning of a log-odds change

Logarithms are useful because they turn multiplication into addition:

Correspondingly, exponentials turn addition back into multiplication:

Suppose log-odds change by some amount $d$:

Exponentiating both sides implies:

Since $e^z$ is odds,

Thus:

is an odds ratio: the factor by which the odds change.

For example, if $d=\ln (2)$, then

A $\ln (2)$ increase in log-odds doubles the odds.

However, doubling odds does not mean doubling probability. The resulting probability depends on the starting probability.

Suppose a workspace starts with:

Its odds are:

Doubling odds gives:

Converting back:

So its probability rises from 1% to about 1.98%, not to 2% exactly and certainly not to 100%.

Now start instead at:

The starting odds are:

Doubling those odds gives $3$, which corresponds to:

The same odds ratio of $2$ produces a change from 60% to 75%.

This distinction will matter when reading logistic-regression results later:

• an odds ratio above $1$ means higher odds;
• an odds ratio below $1$ means lower odds;
• it is not a direct percentage-point change in probability;
• it is also not, by itself, a causal effect.

A compact conversion workflow in Python

For routine work, it helps to calculate all representations together and inspect whether the values make sense.

import numpy as np

p = np.array([0.20, 0.50, 0.80])

odds = p / (1 - p)
log_odds = np.log(odds)

# Invert log-odds with the sigmoid formula
p_recovered = 1 / (1 + np.exp(-log_odds))

for probability, odd, logit, recovered in zip(
    p, odds, log_odds, p_recovered
):
    print(
        f"p={probability:.2f}, "
        f"odds={odd:.3f}, "
        f"log-odds={logit:.3f}, "
        f"recovered p={recovered:.2f}"
    )

The expected output is approximately:

p=0.20, odds=0.250, log-odds=-1.386, recovered p=0.20
p=0.50, odds=1.000, log-odds=0.000, recovered p=0.50
p=0.80, odds=4.000, log-odds=1.386, recovered p=0.80

There is one mathematical boundary condition worth remembering. The logit is undefined at exactly $0$ and $1$:

is not finite, and neither is the logit at $p=1$. In contrast, the sigmoid of any finite $z$ is always strictly inside the interval:

This is one reason the sigmoid is well suited to transforming a finite model score into a probability.

Key takeaways

The five transformations in this lesson form one connected system:

The sigmoid maps an unrestricted linear score to a valid probability. A change in log-odds corresponds to a multiplicative change in odds after exponentiation, but the implied probability change depends on the initial probability.

Next, the course moves to the calculus tools behind learning model parameters: the product rule and chain rule.